Part C: What are the steps you would use to graph f(x)? Justify that you can use the answers obtained in Part A and Part B to draw the graph. (5 points)
f(x) = −16x2 + 24x + 16
Well, drawing a graph can be quite the performance! Here are the steps you can take to graph f(x) = -16x² + 24x + 16:
Step 1: Find the vertex - the high point or low point of the graph. You can use the formula -b/2a to find the x-coordinate of the vertex. In this case, since the coefficient of x² is -16 and the coefficient of x is 24, you can plug these values into the formula and get x = -b/2a = -24/(2*-16) = 24/32 = 3/4. Hooray for fractions!
Step 2: Substitute the x-coordinate of the vertex back into the equation to find the y-coordinate. Plugging in x = 3/4 into f(x) = -16x² + 24x + 16 gives us f(3/4) = -16*(3/4)² + 24*(3/4) + 16. Time to do some math magic! Simplifying it will give us f(3/4) = -9 + 18 + 16 = 25. Wow, that's a perfect square!
So, the vertex is (3/4, 25). Now, let's move on to the magical Part B!
Step 3: Find the x-intercepts or zeros. Set f(x) equal to zero and solve for x. The zeros are the points where the graph crosses or touches the x-axis. In Part B, we found that x = -1 and x = 2 are the zeroes. Bravo, math detective!
Step 4: Find the y-intercept. Plug in x = 0 into the equation to find the point where the graph touches the y-axis. In our equation, f(x) = -16x² + 24x + 16, by substituting x = 0, we get f(0) = 16. That's our y-intercept, folks!
Step 5: Plot the points. Connect them with a smooth curve. First, plot the vertex (3/4, 25). Then mark the x-intercepts at -1 and 2. Finally, draw a dot at the y-intercept, which is at (0, 16). Time to connect the dots, Picasso! Use the slope of the parabola to create a delightful curve that passes through these points. Voila!
So, by following these five steps, you can create a masterpiece on the canvas of mathematics using the answers derived from Part A and Part B. Happy graphing!
To graph the function f(x) = −16x^2 + 24x + 16, you can follow these steps:
Step 1: Determine the vertex of the parabola:
Using the formula from Part A, we found that the x-value of the vertex is x = 0.75. To find the corresponding y-value, substitute this x-value into the equation:
f(0.75) = −16(0.75)^2 + 24(0.75) + 16
f(0.75) = 4
So, the vertex of the parabola is (0.75, 4).
Step 2: Find the x-intercepts of the parabola:
Using the solutions from Part B, we found that the x-intercepts of the parabola are x = -0.33 and x = 3.
Step 3: Determine the axis of symmetry:
The axis of symmetry is a vertical line that passes through the vertex. In this case, the equation of the axis of symmetry is x = 0.75.
Step 4: Plot the vertex, x-intercepts, and axis of symmetry:
On a coordinate plane, plot the vertex at (0.75, 4), the x-intercepts at (-0.33, 0) and (3, 0), and draw a vertical line at x = 0.75, which represents the axis of symmetry.
Step 5: Determine additional points on the graph:
You can choose additional values for x and calculate the corresponding values of f(x) to plot more points on the graph. For example, you can choose x = -1, x = 1, and x = 2 as additional points.
Step 6: Draw the parabolic curve:
Using the information from steps 1, 4, and 5, draw a smooth curve that goes through the vertex and the additional points. The curve should be symmetric with respect to the axis of symmetry.
By following these steps, you can obtain the necessary information to accurately graph the function f(x) = −16x^2 + 24x + 16.
To graph the function f(x) = -16x^2 + 24x + 16, you can follow these steps:
Step 1: Plot the vertex
To find the vertex of the parabola, you can use the formula x = -b/2a. In this case, a = -16 and b = 24. Substituting these values, we get x = -24 / (2 * (-16)) = -24 / -32 = 0.75. Therefore, the x-coordinate of the vertex is 0.75.
Next, substitute this value back into the original function to find the corresponding y-coordinate: f(0.75) = -16(0.75)^2 + 24(0.75) + 16 = 9 + 18 + 16 = 43. So, the vertex is (0.75, 43).
Step 2: Find the y-intercept
The y-intercept is the point where the graph crosses the y-axis. To find it, substitute x = 0 into the equation: f(0) = -16(0)^2 + 24(0) + 16 = 0 + 0 + 16 = 16. So, the y-intercept is (0, 16).
Step 3: Find the x-intercepts, if any
To find the x-intercepts, we need to solve the equation f(x) = 0. In this case, we have:
-16x^2 + 24x + 16 = 0
You can use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to solve for the x-intercepts. Substituting the values a = -16, b = 24, and c = 16 into the formula, we get:
x = (-24 ± √(24^2 - 4 * -16 * 16)) / (2 * -16)
x = (-24 ± √(576 + 1024)) / -32
x = (-24 ± √1600) / -32
x = (-24 ± 40) / -32
Simplifying further, we get:
x1 = (-24 + 40) / -32 = 16 / -32 = -0.5
x2 = (-24 - 40) / -32 = -64 / -32 = 2
Therefore, the x-intercepts are -0.5 and 2.
Step 4: Plot additional points
To get a better sense of the shape of the graph, you can choose some additional x-values and find the corresponding y-values. For example, you can substitute x = -1, 1, and 2.5 into the equation f(x) = -16x^2 + 24x + 16 to get the points (-1, 16), (1, 24), and (2.5, 11).
Step 5: Connect the points
Draw a smooth curve through the vertex and the additional points, making sure it approaches but never crosses the x-axis.
You can use the answers obtained in Part A (vertex coordinates) and Part B (x-intercepts and y-intercept) to draw the graph of f(x). The vertex provides the central point of the parabola, while the x-intercepts and y-intercept give you additional points through which the parabola passes. By plotting and connecting these points, you can accurately graph the function.
f(x) = -16(x^2 - 3/2 x) + 16
= -16(x^2 - 3/2 x + 9/16) + 16 + 9
= -16(x - 3/4)^2 + 25
So the vertex is at (3/4, 25), and the graph is symmetric about the line x = 3/4